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ItemNo Preview AvailableNonlinear, statistical and stochastic dynamics of geophysical fluids with application to state estimation and predictionO'Kane, Terence John ( 2021)Many important results underpinning our understanding of variations in the climate have been achieved by linearizing the deterministic equations of motions or through Markovian and Gaussian assumptions on the statistical representation of the systems probability distribution. With the accrual of several decades of ocean and weather observations of sufficient spatial density and homogeneity to resolve the climate modes of variability, it has become evident that the climate system is highly nonlinear, non-stationary and non-Gaussian with transitions between persistent or metastable regimes ubiquitous across all scales. As a result, our ability to make skilful predictions of the state of the various weather systems and climate teleconnections and their causal relationships requires a foundational understanding of the dynamics and life cycles of instability processes that determine predictability. While state of the art operational numerical weather prediction systems now routinely apply aspects of mathematical methods from dynamical systems to characterise the impact of small-scale growing disturbances (error modes) on the predictability of the large scale quasi-stationary structures of the synoptic weather, these approaches are largely empirical. Ideally, a modern mathematical representation of the dynamics of the Earth’s weather and climate must necessarily be cast in terms of a nonlinear, stochastic and multiscale system. However, the sheer complexity and dimensionality of the problem makes development of a foundational theoretical basis for climate variability and predictability a distinct challenge. With the development of novel data driven stochastic optimization methods, high dimensional weather and climate data can now be reduced to low dimensional systems of stochastic differential equations that retain the essential dynamics but make tractable the extraction of regime behaviours of persistent states in systems where the signal to noise ratio is low. The body of work contained in this thesis represents a sustained effort to advance a deeper understanding of the role of nonlinear processes in geophysical fluids with specific application to the development and analysis of weather, ocean and climate prediction systems. Chapter 1 develops the statistical mechanics and dynamics of inhomogeneous turbulent flows with application to modelling unresolved (subgrid) processes associated with the interactions between eddies and topography, and methods of ensemble numerical weather prediction and data assimilation. Mathematically, this is the foundational problem of strong interactions across scales in systems with a quadratic nonlinearity. In essence, one is required to find tractable and accurate representation of an infinite hierarchy of moment equations. The work presented in this chapter develops theoretical approaches borrowed from modern physics and novel computational methods that achieve this goal. This is one of the most technically and intellectually challenging problems in mathematical physics manifesting in areas as diverse as quantum electro- and chromo-dynamics and plasma physics. The theory is validated against ensemble direct numerical simulations of the primitive equations and is shown to enable deep insights into a range of problems of direct societal relevance and in particular weather prediction and state estimation. Chapter 2 describes the development of weather (p228), mesoscale ocean (p242), cyclone (p258) and climate (p296-399) prediction systems. The ensemble weather prediction system described on pages 228-241 forms the basis of the current operational system at the Australian Bureau of Meteorology. The Climate Analysis Forecast Ensemble (CAFE) system described on pages 296-399 is currently operational at CSIRO and is the first system developed in the Southern Hemisphere to contribute to the official World Meteorological Organization near term climate predictions. These systems implement state of the art data assimilation methods to constrain models to observations to generate initial forecast conditions. For the climate, this requires observations of the atmosphere, ocean, land and sea ice to be assimilated and a large ensemble (order of 100) of model forecast simulations employed to generate covariances between the respective domains. The insights gained from statistical dynamics provides additional insights to tackling these challenges. A severe challenge to the analysis of the aforementioned ensemble prediction systems is the dimensionality of the state estimation and forecast data. To underpin the development of such systems, a deeper understanding of the processes and timescales whereby long-term predictive skill resides in the climate system is required. This is a massive challenge given the short observational record in the ocean and large ensembles of initialized forecasts necessary to identify predictive skill. Despite the huge data sets generated, the number of data instances is far exceeded by the dimensionality of the problem, hence the challenge is a classically “small data” issue. The work in Chapter 3 describes the development and application of novel advanced methods from across applied mathematics (dynamical systems, optimization, numerical methods) applied to better interrogate both model and observational data by extracting, not simply correlations between the respective climate modes of variability, but their (conditionally) causal relationships, including the anthropogenic factors that drive their changing relationships and regime transitions over time. This work reveals a detailed understanding of the abrupt climate regime shift that occurred in the Southern Hemisphere climate during the late 1970s. This change in the large-scale circulation precipitated the prolonged periods of reduced rainfall and heat extremes experienced in recent decades over the Australian continent and changes in the statistics of persistent synoptic scale weather patterns. Low frequency variability in the climate system occurs principally via the coupling of the ocean to the atmosphere. The work in Chapter 4 is primarily concerned with understanding the ocean’s response to the constituent components of the atmospheric forcing. Specifically, this work shows how information is communicated via coherent waves in internal oceanic pathways that are analogous to the storm tracks of the atmosphere. The specific role of baroclinically unstable Rossby waves and density compensated salinity anomalies and the mechanisms by which they are excited are examined revealing the importance of coherence resonance processes whereby the synoptic scale atmosphere excites coherent oceanic disturbances on inter-annual timescales. In the final Chapter 5, a range of approaches to deriving reduced order models of the climate modes of variability are presented. Dynamic Bayesian Networks and directed graph methods are employed to quantify causal relationships between the major climate teleconnections in reanalysis data. A minimal skeleton model specific to the Madden Julian Oscillation is identified based on coherences between Kelvin and Rossby waves from a normal mode decomposition of observations. Finally, stochastic linear inverse models are applied to construct reduced order models of the tropical-extra tropical South Pacific Ocean with the synoptic scale Pacific South American (PSA) mode of tropospheric variability identified as the key stochastic forcing.